# Extremely Weak Supervision Inversion of Multi-physical Properties

Multi-physical inversion plays a critical role in geophysics. It has been widely used to infer various physical properties~(such as velocity and conductivity). Among those inversion problems, some are explicitly governed by partial differential equations~(PDEs), while others are not. Without explicit governing equations, conventional multi-physical inversion techniques will not be feasible and data-driven inversion requires expensive full labels. To overcome this issue, we develop a new data-driven multi-physics inversion technique with extremely weak supervision. Our key finding is that the pseudo labels can be constructed by learning the local relationship among geophysical properties at very sparse well-logging locations. We explore a multi-physics inversion problem from two distinct measurements~(seismic and EM data) to three geophysical properties~(velocity, conductivity, and CO$_2$ saturation). Our results show that we are able to invert for properties without explicit governing equations. Moreover, the label data on three geophysical properties can be significantly reduced by 50 times~(from 100 down to only 2 locations).

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